All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$5.50$ each for teachers and $$2.50$ each for students, and the group paid $$31.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$11.00$ each for teachers and $$10.00$ each for students, and the group paid $$93.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+2.5y = 31.5}$ ${11x+10y = 93}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-11x-5y = -63}$ ${11x+10y = 93}$ Add the top and bottom equations together. $ 5y = 30 $ $ y = \dfrac{30}{5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5.5x+2.5y = 31.5}$ to find $x$ ${5.5x + 2.5}{(6)}{= 31.5}$ $5.5x+15 = 31.5$ $5.5x = 16.5$ $x = \dfrac{16.5}{5.5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {11x+10y = 93}$ and get the same answer for $x$ ${11x + 10}{(6)}{= 93}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.